19399번 - Independent Events 스페셜 저지다국어

Yuuka is interested in $n$ independent events. The probability that the $i$-th event occurs is $p_i$. Yuuka is going to perform $m$ operations, each being one of the following:

• "0 $l_i$ $r_i$": considering only the events from $l_i$ to $r_i$ (both inclusive), find the probability that none of these events occur. As the value may be too small, you need to print the natural logarithm of the probability: if the probability is $p$, print $\ln(p)$.
• "1 $l_i$ $r_i$ $k_i$": for all $l_i \le j \le r_i$, multiply $p_j$ by $k_i$. All events remain independent.

The input contains zero or more test cases, and is terminated by end-of-file. For each test case:

The first line contains two integers $n$ and $m$: the number of events and the number of operations ($1 \le n, m \le 10^5$).

The second line contains $n$ real numbers $p_1, p_2, \dots, p_n$ where $p_i$ is the probability that the $i$-th event occurs ($10^{-5} \le p_i \le 0.1$).

The following $m$ lines provide the descriptions of the operations. The $i$-th line starts with an integer $t_i$: the type of the corresponding operation. If $t_i$ is "0", it is followed by two integers $l_i$ and $r_i$. If $t_i$ is "1", it is followed by two integers $l_i$ and $r_i$, and a real number $k_i$ ($1 \le l_i \le r_i \le n$, $0.0001 \le k_i \le 100$).

Each real number in the input has exactly five digits after the decimal point. Additionally, it is guaranteed that, at every moment, every $p_i$ lies in the interval $[10^{-5}, 0.1]$.

It is guaranteed that neither the sum of all $n$ nor the sum of all $m$ will exceed $10^5$.

For each operation of type "0", output a real number denoting the answer. Your answer will be considered correct if its relative error doesn't exceed $10^{-9}$.